# Speed, momentum and total energy of a muon (SR)

1. Jan 9, 2017

### jasonchiang97

1. The problem statement, all variables and given/known data

A muon has a mass of 106MeV/c2. Calculate the speed, momentum and total energy of a 200MeV muon(a muon with a kinetic energy of 200MeV).
2. Relevant equations
E=γmc
K+mc2=E
γ=1/(1-β)1/2
β=(v/c)2

3. The attempt at a solution
To solve for the speed I plugged E=γmc into K+mc2=E to obtain K/mc2 + 1 = γ. Expanding gamma and rearranging I get 1-(K/mc2+1)-1 = β. But plugging in the numbers on the left hand side gives me a zero as K<<c2 and I think that I went wrong somewhere.

2. Jan 9, 2017

### TSny

The first, third, and fourth equations have errors.
[EDIT: Sorry. Your third and fourth equations are OK if you are defining the symbol β to equal (v/c)2. But it is more common to use β for v/c.]

E ≠ γmc. But your expression for γ looks correct.

Check your work here, I don't think you got the correct expression for $\beta$.

Did you mean to write K << mc2? If so, how did you come to this conclusion based on the information given in the problem?

Last edited: Jan 9, 2017
3. Jan 10, 2017

### jasonchiang97

Sorry line 1 was supposed to be E=γmc2 and sorry I skipped so many steps. I'll write down everything I did.

K=mc2(γ-1)
K/mc2 + 1 = 1/(√1-(v2/c2))
1-(v2/c2)=(K/mc2 + 1)2
1-(K/mc2 + 1)2=(v2/c2)
c2(1-(K/mc2 + 1)2)=v2
√[c2(1-(K/mc2 + 1)2)]=v

did I do anything wrong?

4. Jan 10, 2017

### jasonchiang97

Did you mean to write K << mc2? If so, how did you come to this conclusion based on the information given in the problem?

Well given the numbers, c2 = 9 x 1016 and from the values of K and m given K/mc2 = 0

5. Jan 10, 2017

### TSny

Check going from 2nd to 3rd line

6. Jan 10, 2017

### TSny

Note the units in the value of the mass, m =106 MeV/c2.
So c2 cancels in mc2.

7. Jan 10, 2017

### jasonchiang97

I keep making mistakes as I try to type it on the site sorry and thanks.

8. Jan 10, 2017

### jasonchiang97

Yea I know they cancel I mean when you put in the numbers you basically get 1/∞ for E/mc2

9. Jan 10, 2017

### TSny

What numerical value do you get for mc2 (including units)?

(Your final expression for v in post #7 looks correct.)

Last edited: Jan 10, 2017
10. Jan 11, 2017

### jasonchiang97

ohh I see the c2 cancels out. Thanks!